Dynamics 14th edition by r c hibbeler section 13 5

Apply the equation of motion using normal and tangential... The positive n direction of the normal and tangential coordinates is ____________.. The tangential direction t is tangent to

Today’s Objectives:

Students will be able to:

1 Apply the equation of motion

using normal and tangential

2 The positive n direction of the normal and tangential

coordinates is

A) normal to the tangential component

B) always directed toward the center of curvature

C) normal to the bi-normal component

D) All of the above

1 The “normal” component of the equation of motion is written

as F n=ma n, where F n is referred to as the _

A) impulse B) centripetal force

C) tangential force D) inertia force

READING QUIZ

Race track turns are often banked to reduce the frictional forces required to keep the cars from sliding up to the outer rail at high speeds.

If the car’s maximum velocity and a minimum coefficient of

friction between the tires and track are specified, how can we determine the minimum banking angle () required to prevent the car from sliding up the track?

APPLICATIONS

This picture shows a ride at the amusement park The

hydraulically-powered arms turn at a constant rate, which creates

a centrifugal force on the riders

We need to determine the smallest angular velocity of cars A

and B such that the passengers do not lose contact with their

APPLICATIONS (continued)

Satellites are held in orbit around the earth by using the earth’s gravitational pull as the centripetal force – the force acting to change the direction of the satellite’s velocity.

Knowing the radius of orbit of the satellite, we need to

determine the required speed of the satellite to maintain this orbit What equation governs this situation?

APPLICATIONS (continued)

The tangential direction (t) is tangent to the path, usually set as positive in the direction of motion of the particle.

When a particle moves along a

curved path, it may be more convenient to write the equation

of motion in terms of normal and tangential coordinates

The normal direction (n) always points toward the path’s center

of curvature In a circle, the center of curvature is the center of the circle

NORMAL & TANGENTIAL COORDINATES

(Section 13.5)

Since there is no motion in the binormal (b) direction, we can

also write Fb = 0

This vector equation will be satisfied provided the individual

components on each side of the equation are equal, resulting in the two scalar equations: Ft = mat and Fn = man

Here Ft & Fn are the sums of the force components acting in the t & n directions, respectively

Since the equation of motion is a

The tangential acceleration , at = dv/dt, represents the time rate of

change in the magnitude of the velocity Depending on the direction

The normal acceleration , an = v 2 / , represents the time rate of change

in the direction of the velocity vector Remember, an always acts

directed toward the center of the path.

Recall, if the path of motion is defined

any point can be obtained from

NORMAL AND TANGENTIAL ACCELERATION

• Use n-t coordinates when a particle is moving along a known,

curved path.

acceleration (an) always acts “inward” (the positive n-direction) The tangential acceleration (at) may act in either the positive or negative t direction.

at = dv/dt = v dv/ds an = v 2 / 

SOLVING PROBLEMS WITH n-t COORDINATES

Given:The 10-kg ball has a velocity of

3 m/s when it is at A, along the vertical path

increase in the speed of the ball

1) Since the problem involves a curved path and requires finding the force perpendicular to the path, use n-t

coordinates Draw the ball’s free-body and kinetic diagrams

2) Apply the equation of motion in the n-t directions

Plan:

EXAMPLE

1) The n-t coordinate system can

be established on the ball at

Point A, thus at an angle of °

Draw the free-body and kinetic

diagrams of the ball

2) Apply the equations of motion in the n-t directions.

(a) Fn = man  T – W sin ° = m an

EXAMPLE (continued)

2 A 20 lb block is moving along a smooth surface If the

normal force on the surface at A is 10 lb, the velocity is

1 A 10 kg sack slides down a smooth surface If the normal

force at the flat spot on the surface, A, is 98.1 N () , the radius of curvature is

A) 0.2 m B) 0.4 m

C) 1.0 m D) None of the above A

v=2m/s

Given:The boy has a weight of 60 lb

At the instant  = 60, the boy’s

center of mass G experiences a speed v = 15 ft/s.

supporting cords of the swing and the rate of increase in his speed at this instant

1) Use n-t coordinates and treat the boy as a particle Draw the free-body and kinetic diagrams

2) Apply the equation of motion in the n-t directions

Plan:

GROUP PROBLEM SOLVING I

1) The n-t coordinate system can

be established on the boy at

angle ° Approximating the

boy as a particle, the free-body

and kinetic diagrams can be

GROUP PROBLEM SOLVING I (continued)

1) Treat the car as a particle Draw its free-body and kinetic diagrams.

2) Apply the equations of motion in the n-t directions.3) Use calculus to determine the slope and radius of curvature of the path at point A

a hill with the shape of a parabola When the car is at point A, its v = 9 m/s and

a = 3 m/s2 (Neglect the size

of the car.)

exerted on the road at point A by the car

Plan:

GROUP PROBLEM SOLVING II

W = mg = weight of car

N = resultant normal force on road

1) The n-t coordinate system can

be established on the car at

point A Treat the car as a

particle and draw the

free-body and kinetic diagrams:

tn

W

=

GROUP PROBLEM SOLVING II (continued)

2) Apply the equations of motion in the n-t directions:

From Eq (1): N = 7848 cos  – 64800 / 

1 The tangential acceleration of an object

A) represents the rate of change of the velocity vector’s direction

B) represents the rate of change in the magnitude of the velocity

C) is a function of the radius of curvature

D) Both B and C

2 The block has a mass of 20 kg and a speed of

v = 30 m/s at the instant it is at its lowest point

Determine the tension in the cord at this instant

End of the Lecture

Let Learning Continue